Eigenfunction expansions for the Schrödinger equation with inverse-square potential

Abstract

We consider the one-dimensional Schrödinger equation $-f"+q_{\kappa }f=Ef$ on the positive half-axis with the potential $q_{\kappa }(r)=({\kappa }^2-1/4)r^{-2}$. For each complex number $\vartheta$, we construct a solution $u_{\vartheta }^{\kappa }(E)$ of this equation that is analytic in $\kappa$ in a complex neighborhood of the interval $(-1,1)$ and, in particular, at the "singular" point $\kappa =0$. For $-1<\kappa <1$ and real $\vartheta$, the solutions $u_{\vartheta }^{\kappa }(E)$ determine a unitary eigenfunction expansion operator $U_{\kappa ,\vartheta }:L_2(0,\infty )\to L_2(\mathbb{R},{\mathcal{V}}_{\kappa ,\vartheta })$, where ${\mathcal{V}}_{\kappa ,\vartheta }$ is a positive measure on $\mathbb{R}$. We show that every self-adjoint realization of the formal differential expression $-{\partial }_r^2+q_{\kappa }(r)$ for the Hamiltonian is diagonalized by the operator $U_{\kappa ,\vartheta }$ for some $\vartheta \in \mathbb{R}$. Using suitable singular Titchmarsh-Weyl $m$-functions, we explicitly find the measures ${\mathcal{V}}_{\kappa ,\vartheta }$ and prove their continuity in $\kappa$ and $\vartheta$.

20 pages, the contribution to a special issue of Theoretical and Mathematical Physics dedicated to I.V. Tyutin on the occasion of his 75th birthday, final version

References in corpus (1)

• Reduction by symmetries in singular quantum-mechanical problems: general scheme and application to Aharonov-Bohm model

Cited by in corpus (3)

• A Gronwall-type Trigonometric Inequality
• Reduction by symmetries in singular quantum-mechanical problems: general scheme and application to Aharonov-Bohm model
• Coupling constant dependence for the Schrödinger equation with an inverse-square potential